Is there a way to export the color palette (paint mode)?
Import palette from another blend

When you'd like to save a palette using standard features press New button in the palettes dropdown list. New palette will be saved and used instead of default one:



When some changes are made so it to contain desired hues, save the file and open new one. Select option to link or append from file, navigate to that saved file and search for Palettes folder:



You will see there as many palettes as you created in original file.

Paint Palette addon

It's possible to use Paint Palette addon for that, which is bundled with Blender. Addon will save your palette in .gpl format, so you should be able to share saved palette in GIMP (or import that in Blender using this addon):




Press "+" button located leftwards to add a color to palette.
Press "Browse" button to select folder where you'd like to save and click  "Accept" there.
Press "+" button in the top right to save preset using selected path.